title: Beyond simple maps - Integrating space and time with Bayesian models author: - “Corey S. Sparks, Ph.D.” institute: - “Univerity of Texas at San Antonio - Department of Demography” - https://hcap.utsa.edu/demography date: “July 11, 2022” subtitle: Summer at Census Research Seminar output: beamer_presentation
Presentation Structure
- Spatial and temporal demography
- Data sources
- Modeling strategies
- Empirical analysis of Florida mortality rates
- Results & visualizations
- Wrap up
Beyond maps…
.center[
]
class: center, inverse
.center[
]
class: center, inverse
.center[
]
Spatial Demography
- “Putting people into place” (Entwisle, 2007)
- Need to think about:
- Context
- Dynamics
- Processes
| - Macro - demography (Voss, 2007) + Places as observations + Pre - 1960’s + Ecological inference |
|---|
| - Micro - demography + People as observations + Social theory + Individual choices |
- Multilevel - demography
- People in places
- Interaction between context and behavior
Space & Time
- Future
directions in spatial demography report
- Most participants listed time or temporal data as integral to the future of the field
| - Time allows for dynamics of humans and environment + Snap shots/cross sections tell us nothing of this |
|---|
| - Data management + Combining and merging data |
- Analysis/methods
- Problems with space
- Problems with time
| - Advantages + Rich, dynamic contexts + Policy relevance of timely, prospective analysis |
|---|
| - Census/ACS |
- DHS
| - IPUMS |
|---|
| - International agencies |
- Various administrative orgs.
- State government
- Private companies/Nonprofits
How to combine these things?
- Geocodes are essential
- Limitation for many surveys
| - Caveats - Levels of geography + The evil tracts - MAUP - Changing boundaries |
|---|
| - You can basically get these data from the CDC Wonder website - Suppresses counts where the number of deaths is less than 10 - Rates are labeled as “unreliable” when the rate is calculated with a numerator of 20 or less + Big problem for small population counties + Still a problem for large population counties! |
- Restricted use data allows access to ALL data
Data example
| County | Year | Race-Sex | Rate |
|---|---|---|---|
| 12073 | 1980 | White Female | 7.238632 |
| 12073 | 1980 | Black Female | 8.958174 |
| 12073 | 1980 | White Male | 11.840842 |
| 12073 | 1980 | Black Male | 15.907688 |
| 12073 | 1981 | White Female | 7.383039 |
| 12073 | 1981 | Black Female | 9.379846 |
| 12073 | 1981 | White Male | 10.518428 |
| 12073 | 1981 | Black Male | 16.626825 |
| 12073 | 1982 | White Female | 7.370335 |
| 12073 | 1982 | Black Female | 8.695655 |
| 12073 | 1982 | White Male | 11.902308 |
| 12073 | 1982 | Black Male | 12.149819 |
County specific temporal trends 1980 - 2010
Florida Example
- n = 67 counties \(*\) 31 years \(*\) 2 Races \(*\) 2 Sexes = 8,308
Methods - Bayesian Hierarchical models
- Example case of Florida counties
- Examine county-specific time trends in Black/White mortality rates
- I specify a Bayesian Hierarchical model for the age-standardized mortality rate
- Controls for sex and county SES
- Spatial correlation in overall rate \(u_j\)
- Time varying Black/white disparity parameter \(\nu_{t2}\)
- Spatially varying Black/White disparity parameter \(\gamma_j\)
\[ \begin{aligned} \operatorname{y}_{ij} &\sim N\left( \mu, \tau_y \right) \\ & \mu_{ij} = \beta_{0} + x'\beta +\gamma_j*Black + u_j +\nu_{t1} + \nu_{t2}* Black \\ & \gamma_j \sim \text{CAR}(\bar \gamma_j, \tau_{\gamma}/n_j) \\ & u_j \sim \text{CAR}(\bar u_j, \tau_u /n_j)\\ & \nu_{t2} \sim RW1(time)\\ & \nu_{t1} \sim N(0, \tau_t) \\ \end{aligned} \]
Methods - Bayesian Hierearchical models
- This type of model is commonly used in epidemiology and public health
- Various types of data likelihoods may be used
- Need to get at:
*\[p(\theta|y) \propto p(y|\theta)p(\theta)\]
- Traditionally, we would get \(p(\theta|y)\) by:
- either figuring out what the full conditionals for all our model parameters are (hard)
- Use some form of MCMC to arrive at the posterior marginal distributions for our parameters (time consuming)
Methods - INLA approach
- Integrated Nested Laplace Approximation - Rue, Martino & Chopin (2009)
- One of several techniques that approximate the marginal and
conditional posterior densities
- Laplace, PQL, E-M, Variational Bayes
- Assumes all random effects in the model are latent, zero-mean
Gaussian random field, \(x\) with some
precision matrix
- The precision matrix depends on a small set of hyperparameters
- Attempts to construct a joint Gaussian approximation for \(p(x | \theta, y)\)
- where \(\theta\) is a small subset of hyper-parameters
Methods - INLA approach
Apply these approximations to arrive at:
\(\tilde{\pi}(x_i | y) = \int \tilde{\pi}(x_i |\theta, y)\tilde{\pi}(\theta| y) d\theta\)
\(\tilde{\pi}(\theta_j | y) = \int \tilde{\pi}(\theta| y) d\theta_{-j}\)
where each \(\tilde{\pi}(. |.)\) is an approximated conditional density of its parameters
Approximations to \(\pi(x_i | y)\) are computed by approximating both \(\pi(\theta| y)\) and \(\pi(x_i| \theta, y)\) using numerical integration to integrate out the nuisance parameters.
- This is possible if the dimension of \(\theta\) is small.
Approximations to \(\tilde{\pi}(\theta|y)\) are based on the Laplace appoximation of the marginal posterior density for \(\pi(x,\theta|y)\)
Their approach relies on numerical integration of the posterior of the latent field, as opposed to a pure Gaussian approximation of it
INLA in R
library(INLA)
std_rate~male+black+scale(lths)+
f(year2, model = "rw1",constr = T, scale.model = T)+
nonparametric time trend
f(struct, model="besag", graph="cl_graph", constr = T, scale.model = T)+
spatial correlation
f(year3, bl2, model="iid")+ time -
disparity
f(struct2, bl2, model="besag", graph="cl_graph", constr = T, scale.model = T)
spatial disparity
Results
- Time trend in Black/white Mortality
County time trends
Highlighed trends
Spatial trend
Spatial disparity
Discussion
- We see that, while there is a persistence of the gap in black-white
mortality:
- The mortality gap appears to be fairly consistent over time
- In some areas, the mortality difference are decreasing
- Results point to higher disparities in several notable Florida rural areas
- Spatio-temporal modeling allows for the incorporation of dynamics that cross-sectional models cannot
Low Response Score Outcome
- INLA model for Low Response Score metric
- Considered both an unstructured and spatially structured random effect model
- Modeled LRS as Gaussian, considering how it is constructed
- Besag, York and Mollie specification for tract level heterogeneity
\[ \begin{aligned} \operatorname{y}_{i} &\sim \text{Normal}\left( \mu_i, \tau_y \right) \\ & \mu_{i} = \beta_{0} + u_i + v_i \\ & u_i \sim \text{CAR}(\bar u_i, \tau_{u}/n_j) \\ & u_i \sim \text{Normal}(\bar 0, \tau_{v}/n_j) \\ \end{aligned} \]
Low Response Score in Texas
Spatial correlation random effect
IID random effect
- Spatial model fits much better than non-spatial model using WAIC
- Suggests model should take into account spatial structure in LRS underlying data
- Larger suggestion is that spatial correlation needs to be included in the underlying construction of the LRS
More discussion
- INLA allows for rapid deployment of Bayesian statistical models with
latent Gaussian random effects
- Faster and generally as accurate as MCMC
- Potentially an attractive solution for problems where large data/complex models may make MCMC less desirable
Thank you!
Slides created via the R package xaringan
All talk materials available at my Github page
R-INLA examples available at my Rpubs page